Peak Sidelobe Level and Peak Crosscorrelation of Golay–Rudin–Shapiro Sequences

Sequences with low aperiodic autocorrelation and crosscorrelation are used in communications and remote sensing. Golay and Shapiro independently devised a recursive construction that produces families of complementary pairs of binary sequences. In the simplest case, the construction produces the Rudin–Shapiro sequences, and in general it produces what we call Golay–Rudin–Shapiro sequences. Calculations by Littlewood show that the Rudin–Shapiro sequences have low mean square autocorrelation. A sequence’s peak sidelobe level is its largest magnitude of autocorrelation over all nonzero shifts. Høholdt, Jensen, and Justesen showed that there is some undetermined positive constant
$A$
such that the peak sidelobe level of a Rudin–Shapiro sequence of length
$2^{n}$
is bounded above by
$A(1.842626\ldots)^{n}$
, where
$1.842626\ldots $
is the positive real root of
$X^{4}-3 X-6$
. We show that the peak sidelobe level is bounded above by
$5(1.658967\ldots)^{n-4}$
, where
$1.658967\ldots $
is the real root of
$X^{3}+X^{2}-2 X-4$
. Any exponential bound with lower base will fail to be true for almost all
$n$
, and any bound with the same base but a lower constant prefactor will fail to be true for at least one
$n$
. We provide a similar bound on the peak crosscorrelation (largest magnitude of crosscorrelation over all shifts) between the sequences in each Rudin–Shapiro pair. The methods that we use generalize to all families of complementary pairs produced by the Golay–Rudin–Shapiro recursion, for which we obtain bounds on the peak sidelobe level and peak crosscorrelation with the same exponential growth rate as we obtain for the original Rudin–Shapiro sequences.